aircraft control Lecture 7

7/29/2019 aircraft control Lecture 7

1/15

16.333: Lecture# 7ApproximateLongitudinalDynamicsModels

Acouplemorestabilityderivatives Given mode shapes found identify simpler models that capture the main re-

sponses


2/15

Fall2004 16.33361MoreStabilityDerivatives

Recallfrom62thatthederivativestabilityderivativetermsZw and

Mw endedupontheLHSasmodificationstothenormalmassandinertiatermsThese are the apparentmass effects someof the surroundingdisplacedairisentrainedandmoveswiththeaircraft

AccelerationderivativesquantifythiseffectSignificantforblimps,lesssoforaircraft.

Maineffect: rateofchangeofthenormalvelocityw causesatransientinthedownwashfromthewingthatcreatesachangeintheangleofattackofthetailsometimelaterDownwashLageffect

IfaircraftflyingatU0,willtakeapproximatelyt=lt/U0 toreachthetail.Instantaneousdownwashatthetail(t) isduetothewingattimett.

(t)=

(tt)

Taylorseriesexpansion(tt)(t)t

Notethat(t) =t. Change inthetailAOAcanbecom-putedas

d d lt(t)= t= =t

d d U0


3/15

Fall2004 16.33362 For the tail,we have that the lift increment due to the change in

downwashisd

ltCLt =CLtt =CLtd U0

Thechangeinliftforceisthen1

Lt = (U02)tStCLt2IntermsoftheZ-forcecoefficient

Lt

St

St

d ltCZ = 1 = CLt = CLt U02S S S d U 02 c/(2U0)tonondimensionalizetime,sotheappropriatestabil-Weuse

itycoefficientformis(noteuseCz tobegeneral,butwearelookingatCz frombefore):

CZ 2U0 CZ= =CZ 0( c/2U0) c 02U0St lt d

= c S U0CLtd

d= 2VHCLtd

ThepitchingmomentduetotheliftincrementisMcg = ltLt

1(U20)tStCLt1CMcg = lt2 U02Sc2

d lt=

VHCLt =

VHCLtd U0


4/15

Fall2004 16.33363

SothatCM 2U0 CM

=

=CM 0( c/2U0) c 0d lt 2U0

= VHCLtd U0 cdlt

= 2VHCLtdclt

CZ c Similarly,pitchingmotionoftheaircraftchangestheAOAofthetail.

Nosepitchupatrateq,increasesapparentdownwardsvelocityoftailbyqlt,changingtheAOAby

qltt =

U0whichchangestheliftatthetail(andthemomentaboutthecg).

Followingsameanalysisasabove: LiftincrementLt = CLtqltU0

12(U20)tSt

CZ = Lt12(U20)S =

StSCLt

qltU0

CZq CZ(qc/2U0) 0 =

2U0c

CZq 0 =

2U0c

ltU0

StSCLt

= 2VHCLtCanalsoshowthat

ltCMq = CZq c


5/15

Fall2004 16.33364ApproximateAircraftDynamicModels

It isoftengoodtodevelopsimplermodelsofthe fullsetofaircraftdynamics.

Provides insightson the roleof theaerodynamicparametersonthefrequencyanddampingofthetwomodes.

Usefulforthecontroldesignworkaswell

Basicapproachistorecognizethatthemodeshaveveryseparatesetsofstatesthatparticipateintheresponse.

ShortPeriodprimarilyandw inthesamephase.Theuandq responseisverysmall.

Phugoidprimarilyandu,and lagsbyabout90.Thewandq responseisverysmall.

Fullequationsfrombefore: Xu

u m

w Zuw

[Mu+Zu]

=

mZ

q

Iyy0

XwmZw

mZw[Mw+Zw]

Iyy0

0Zq+mU0mZw

[Mq+(Zq+mU0)]Iyy1

gcos0 u Xcmgsin0mZ w Zc

mgsin0w

q + McIyy0 0


6/15

Fall2004 16.33365 FortheShortPeriodapproximation,

1.Since u

0 in thismode, then u

0 and can eliminate theX-forceequation. Zq+mU0 mgsin0

mZZw Zcw wmZw mZw[Mq+(Zq+mU0)]

w = +[Mw+Zw] Mcmgsin0Iyyq qIyy Iyy 00 1 02.TypicallyfindthatZw

mandZq

mU0.Checkfor747:

Zw =1909m=2.8866105Zq =4.5105mU0 =6.8107

Mw Mw=

mZw m Zw

U0 gsin0 Zcw wm Mw+ZwMw Mq+(mU0)Mwm = +Mcmgsin0MwIyyq m qIyy Iyy m 00 1 0

3.Set0 =0andremovefromthemodel(itcanbederivedfromq)

Withtheseapproximations,thelongitudinaldynamicsreducetoxsp =Aspxsp+Bspe

wheree istheelevatorinput,andw Zw/m U0xsp = q , Asp = I1(Mw +MwZw/m) I1(Mq +MwU0)yy yy

Ze/mBsp = I1(Me +MwZe/m)yy


7/15

Fall2004 16.33366 Characteristic equation for this system: s2 +2spsps+2 = 0,sp

wherethefullapproximationgives:Zw Mq Mw

2spsp = + + U0m Iyy Iyy

2 ZwMq U0Mw=sp mIyy Iyy

Givenapproximatemagnitudeofthederivativesforatypicalaircraft,candevelopacoarseapproximate:

2spsp MqIyy sp Mq2 1U0MwIyy

2 U0Mw sp sp U0MwIyyIyyNumericalvaluesfor747

Frequency Dampingrad/sec

Fullmodel 0.962 0.387FullApproximate 0.963 0.385CoarseApproximate 0.906 0.187

Bothapproximationsgivethefrequencywell,butfullapproximationgivesamuchbetterdampingestimate

Approximations showed that shortperiodmode frequency is deter-minedbyMw measureoftheaerodynamicstiffnessinpitch.Sign of Mw negative if cg sufficient far forward changes sign(mode goes unstable)when cg at the stick fixed neutral point.FollowsfromdiscussionofCM (see211)


8/15

Fall2004 16.33367 ForthePhugoidapproximation,startagainwith:

Xu Xw 0 gcos0m m mgsin0 Xcu u

Zq+mU0Zu Zw

Z

cMc

wq wqmZw mZw mZw mZw[Mq+(Zq+mU0)] += [Mu+Zu] [Mw+Zw] mgsin0IyyIyy Iyy Iyy 00 0 1 0

1.Changestowandqareverysmallcomparedtou,sowecanSetw 0andq 0Set0 =0

Xu Xw 0 gm m Xcu u

Zq+mU0Zu Zw

00

ZcMc

00 wqmZw mZw mZw[Mq+(Zq+mU0)] += [Mu+Zu] [Mw+Zw]

Iyy Iyy Iyy 00 0 1 02.Usewhat is leftof the Z-equation to show thatwith theseap-proximations(elevatorinputs) ZeZuZq+mU0Zw

mZw mZw mZwmZw

w

u e=

[Mw+Zw] [Mq+(Zq+mU0)] [Me+Ze][Mu+Zu]qIyyIyy Iyy Iyy

3.Use(Zw msoMw )and(ZqmU0)sothat:mZw mU0 w

Mw +Zw Mw [Mq +U0Mw] qm

Zu Ze= Mu+Zu Mw u Mw m Me +Ze m e


9/15

Fall2004 16.333684.Solvetoshowthat

wq =

mU0Mu

ZuMq

ZwMqmU0MwZuMwZwMu

u+mU0Me

ZeMq

ZwMqmU0MwZeMwZwMe

eZwMqmU0Mw ZwMqmU0Mw

5.Substituteintothereducedequationstogetfullapproximation:mU M Z M 0 u u qZwMqmU0Mw

Xu Xw+ gu

m m u=

ZuMwZwMu 0ZwMqmU0Mw Xe Xw mU0MeZeMq+m m ZwMqmU0Mw

e+

ZeMwZwMeZwMq

mU0Mw

6.Stillabitcomplicated.Typicallygetthat(1.4:4)|MuZw| |MwZu|(1:0.13)|MwU0m| |MqZw|

MuXw/Mw|

Xu small|

7.With theseapproximations, the longitudinaldynamics reduce tothecoarseapproximation

xph =Aphxph+Bphewheree istheelevatorinput.


10/15

Fall2004 16.33369And

u

xph = Aph = Xu

g

m

Zu0

mU0

Bph =

MeMwXe Xwm

Zw+

Ze Mw Me

mU0

8.Which

gives

2phph = Xu/m

2 gZu=ph mU0

Numericalvaluesfor747Frequency Dampingrad/sec

Fullmodel 0.0673 0.0489FullApproximate 0.0670 0.0419CoarseApproximate 0.0611 0.0561


11/15

Fall2004 16.333610 Furtherinsights: recallthat

U0 Z U0 L= + 2CL0)QS u 0 QS u 0 (CLu

M2=

1M2CL0 2CL0 2CL0so

Z UoS 2mgZu = (2CL0) = u 0 2 U0

Then

mg2ph = gZu =

mU02mU0

g= 2

U0whichisexactlywhatLanchestersapproximationgave 2 gU0Notethat

X UoSXu = (2CD0) = UoSCD0u 0 2

and2mg= U2SCL0o

soXu XuU0ph =

2mph = 22mg1 U2SCD0o=

2 U2SCL0o1 CD0=

2 CL0sothedampingratiooftheapproximatephugoidmodeisinverselyproportionaltothelifttodragratio.


12/15

Fall2004 16.333611

102

10

1

10

0

10

0

10

1

10

2

10

3

10

4

|Gude

|

F

req(rad/sec)

Transferfunction

frome

levatortoflightvariables

102

10

1

10

0

10

1

10

0

10

1

10

2

10

3

10

4

|Gde

|

Freq

(rad/sec)

102

10

1

10

0

250

200

150

100

500

50

argGude

F

req

(rad/sec)

102

10

1

10

0

350

300

250

200

150

100

500

argGde

Freq

(rad/sec)

FreqComparisonfromelevator(PhugoidModel)B747atM=0.8.BlueFullmodel,BlackFullapproximatemodel,MagentaCoarseapproximatemodel


13/15

Fall2004 16.333612

102

10

1

10

0

10

1

10

0

10

1

10

2

|G

de

|

F

req(rad/sec)

Transferfunction

frome

levatortoflightvariables

102

10

1

10

0

10

1

10

0

10

1

10

2

|Gde

|

Freq

(rad/sec)

102

10

1

10

0

050

100

150

200

250

300

argGude

F

req

(rad/sec)

102

10

1

10

0

350

300

250

200

150

100

500

argGde

Freq

(rad/sec)

FreqComparisonfromelevator(ShortPeriodModel)B747atM=0.8.BlueFullmodel,MagentaApprox-imatemodel


14/15

Fall2004 16.333613Summary

Approximatelongitudinalmodelsarefairlyaccurate

Indicate that theaircraft responsesaremainlydeterminedby thesestabilityderivatives:

Property StabilityderivativeDampingoftheshortperiod MqFrequencyoftheshortperiod MwDampingofthePhugoid XuFrequencyofthePhugoid Zu


15/15

Fall2004 16.333614 Givenachange in,expectchanges inuaswell. Thesewillboth

impact the lift and drag of the aircraft, requiring thatwe re-trimthrottlesettingtomaintainwhateveraspectsoftheflightconditionmighthavechanged(otherthantheoneswewantedtochange).Wehave:

L Lu L u=D Du D

ButtomaintainL=W,wantL=0,sou= Lu L

GivingD= LDu+D Lu2CL0CD = CL D =QSCDeAR

L =QSCLQS

Du = (2CD0) (416)U0QS

Lu = (2CL0) (417)U0CL 2CD0D = QS +CD 2CL0/U0 U0

QS 2C2L0=CL0

CD0 +eAR CL(T0+T)(D0+D) Dtan = = L0+L L0

CD0 2CL0 CL=CL0

eAR CL0

For 747 (Reid 165 andNelson 416), AR = 7.14, so eAR 18,CL0 = 0.654 CD0 = 0.043, CL = 5.5, for a =

0.0185rad

(67) = 0.0006rad. This is the opposite sign to the linearsimulationresults,buttheyarebothverysmallnumbers.

aircraft control Lecture 7

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